LMFDB - Embedded newform 1600.2.d.g.801.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(801,1600)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1600.801");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.7760643234$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 320)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			801.1
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	1600.801
Dual form			1600.2.d.g.801.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.47214	−1.69031	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	−4.47214	−1.69031	−0.845154	+	0.534522i	$0.820491\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	− 2.00000i	− 0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$	− 2.00000i	− 0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$	0	0
$13$	4.47214i	1.24035i	0.784465	+	0.620174i	$0.212938\pi$
$13$	4.47214i	1.24035i	−0.784465	+	0.620174i	$0.787062\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	− 6.00000i	− 1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$	− 6.00000i	− 1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.47214	0.932505	0.466252	−	0.884652i	$-0.345604\pi$
$23$	4.47214	0.932505	0.466252	+	0.884652i	$0.345604\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 4.47214i	− 0.735215i	−0.929981	−	0.367607i	$-0.880177\pi$
$37$	− 4.47214i	− 0.735215i	0.929981	−	0.367607i	$-0.119823\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	13.4164	1.95698	0.978492	−	0.206284i	$-0.0661372\pi$
$47$	13.4164	1.95698	0.978492	+	0.206284i	$0.0661372\pi$
$48$	0	0
$49$	13.0000	1.85714
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 13.4164i	− 1.84289i	−0.388514	−	0.921443i	$-0.627012\pi$
$53$	− 13.4164i	− 1.84289i	0.388514	−	0.921443i	$-0.372988\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 14.0000i	− 1.82264i	−0.411693	−	0.911322i	$-0.635063\pi$
$59$	− 14.0000i	− 1.82264i	0.411693	−	0.911322i	$-0.364937\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−13.4164	−1.69031
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.94427i	1.01929i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	− 20.0000i	− 2.09657i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	− 6.00000i	− 0.603023i
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	4.47214	0.440653	0.220326	−	0.975426i	$-0.429288\pi$
$103$	4.47214	0.440653	0.220326	+	0.975426i	$0.429288\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	13.4164i	1.24035i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−22.3607	−1.98419	−0.992095	−	0.125491i	$-0.959949\pi$
$127$	−22.3607	−1.98419	−0.992095	+	0.125491i	$0.959949\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	22.0000i	1.92215i	0.276289	+	0.961074i	$0.410895\pi$
$131$	22.0000i	1.92215i	−0.276289	+	0.961074i	$0.589105\pi$
$132$	0	0
$133$	26.8328i	2.32670i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	− 14.0000i	− 1.18746i	−0.804663	−	0.593732i	$-0.797654\pi$
$139$	− 14.0000i	− 1.18746i	0.804663	−	0.593732i	$-0.202346\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.94427	0.747958
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 22.3607i	− 1.78458i	−0.451466	−	0.892288i	$-0.649099\pi$
$157$	− 22.3607i	− 1.78458i	0.451466	−	0.892288i	$-0.350901\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−20.0000	−1.57622
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.47214	−0.346064	−0.173032	−	0.984916i	$-0.555356\pi$
$167$	−4.47214	−0.346064	−0.173032	+	0.984916i	$0.555356\pi$
$168$	0	0
$169$	−7.00000	−0.538462
$170$	0	0
$171$	− 18.0000i	− 1.37649i
$172$	0	0
$173$	− 13.4164i	− 1.02003i	−0.860165	−	0.510015i	$-0.829640\pi$
$173$	− 13.4164i	− 1.02003i	0.860165	−	0.510015i	$-0.170360\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	26.0000i	1.94333i	0.236360	+	0.971666i	$0.424046\pi$
$179$	26.0000i	1.94333i	−0.236360	+	0.971666i	$0.575954\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 22.3607i	− 1.59313i	−0.604551	−	0.796566i	$-0.706648\pi$
$197$	− 22.3607i	− 1.59313i	0.604551	−	0.796566i	$-0.293352\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	13.4164	0.932505
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	22.0000i	1.51454i	0.653101	+	0.757271i	$0.273468\pi$
$211$	22.0000i	1.51454i	−0.653101	+	0.757271i	$0.726532\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	22.3607	1.49738	0.748691	−	0.662919i	$-0.230683\pi$
$223$	22.3607	1.49738	0.748691	+	0.662919i	$0.230683\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	26.8328	1.70733
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 2.00000i	− 0.126239i	−0.998006	−	0.0631194i	$-0.979895\pi$
$251$	− 2.00000i	− 0.126239i	0.998006	−	0.0631194i	$-0.0201049\pi$
$252$	0	0
$253$	− 8.94427i	− 0.562322i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	20.0000i	1.24274i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−31.3050	−1.93035	−0.965173	−	0.261612i	$-0.915746\pi$
$263$	−31.3050	−1.93035	−0.965173	+	0.261612i	$0.915746\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	31.3050i	1.88093i	0.339887	+	0.940466i	$0.389611\pi$
$277$	31.3050i	1.88093i	−0.339887	+	0.940466i	$0.610389\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.94427	−0.527964
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.47214i	0.261265i	0.991431	+	0.130632i	$0.0417008\pi$
$293$	4.47214i	0.261265i	−0.991431	+	0.130632i	$0.958299\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	20.0000i	1.15663i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	31.3050i	1.75826i	0.476581	+	0.879131i	$0.341876\pi$
$317$	31.3050i	1.75826i	−0.476581	+	0.879131i	$0.658124\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−60.0000	−3.30791
$330$	0	0
$331$	− 18.0000i	− 0.989369i	−0.869072	−	0.494685i	$-0.835284\pi$
$331$	− 18.0000i	− 0.989369i	0.869072	−	0.494685i	$-0.164716\pi$
$332$	0	0
$333$	− 13.4164i	− 0.735215i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−26.8328	−1.44884
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−22.3607	−1.16722	−0.583609	−	0.812035i	$-0.698360\pi$
$367$	−22.3607	−1.16722	−0.583609	+	0.812035i	$0.698360\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	60.0000i	3.11504i
$372$	0	0
$373$	− 31.3050i	− 1.62091i	−0.585802	−	0.810454i	$-0.699220\pi$
$373$	− 31.3050i	− 1.62091i	0.585802	−	0.810454i	$-0.300780\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	34.0000i	1.74646i	0.487306	+	0.873231i	$0.337980\pi$
$379$	34.0000i	1.74646i	−0.487306	+	0.873231i	$0.662020\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−13.4164	−0.685546	−0.342773	−	0.939418i	$-0.611366\pi$
$383$	−13.4164	−0.685546	−0.342773	+	0.939418i	$0.611366\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 4.47214i	− 0.224450i	−0.993683	−	0.112225i	$-0.964202\pi$
$397$	− 4.47214i	− 0.224450i	0.993683	−	0.112225i	$-0.0357978\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−38.0000	−1.89763	−0.948815	−	0.315833i	$-0.897716\pi$
$401$	−38.0000	−1.89763	−0.948815	+	0.315833i	$0.897716\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.94427	−0.443351
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	62.6099i	3.08083i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	26.0000i	1.27018i	0.772437	+	0.635092i	$0.219038\pi$
$419$	26.0000i	1.27018i	−0.772437	+	0.635092i	$0.780962\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	40.2492	1.95698
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 26.8328i	− 1.28359i
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	39.0000	1.85714
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	0	0
$451$	− 4.00000i	− 0.188353i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	22.3607	1.03919	0.519594	−	0.854413i	$-0.326083\pi$
$463$	22.3607	1.03919	0.519594	+	0.854413i	$0.326083\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 40.2492i	− 1.84289i
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	31.3050	1.41856	0.709281	−	0.704925i	$-0.249020\pi$
$487$	31.3050	1.41856	0.709281	+	0.704925i	$0.249020\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 2.00000i	− 0.0902587i	−0.998981	−	0.0451294i	$-0.985630\pi$
$491$	− 2.00000i	− 0.0902587i	0.998981	−	0.0451294i	$-0.0143700\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 6.00000i	− 0.268597i	−0.990941	−	0.134298i	$-0.957122\pi$
$499$	− 6.00000i	− 0.268597i	0.990941	−	0.134298i	$-0.0428781\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	40.2492	1.79462	0.897312	−	0.441397i	$-0.145517\pi$
$503$	40.2492	1.79462	0.897312	+	0.441397i	$0.145517\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 26.8328i	− 1.18011i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−38.0000	−1.66481	−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000	−1.66481	−0.832405	+	0.554168i	$0.813037\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−3.00000	−0.130435
$530$	0	0
$531$	− 42.0000i	− 1.82264i
$532$	0	0
$533$	8.94427i	0.387419i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 26.0000i	− 1.11990i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.4164i	0.568471i	0.958754	+	0.284236i	$0.0917398\pi$
$557$	13.4164i	0.568471i	−0.958754	+	0.284236i	$0.908260\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−40.2492	−1.69031
$568$	0	0
$569$	46.0000	1.92842	0.964210	−	0.265139i	$-0.0854179\pi$
$569$	46.0000	1.92842	0.964210	+	0.265139i	$0.0854179\pi$
$570$	0	0
$571$	− 18.0000i	− 0.753277i	−0.926360	−	0.376638i	$-0.877080\pi$
$571$	− 18.0000i	− 0.753277i	0.926360	−	0.376638i	$-0.122920\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−26.8328	−1.11130
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	49.1935	1.99670	0.998351	−	0.0574012i	$-0.0182814\pi$
$607$	49.1935	1.99670	0.998351	+	0.0574012i	$0.0182814\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	60.0000i	2.42734i
$612$	0	0
$613$	− 49.1935i	− 1.98691i	−0.114239	−	0.993453i	$-0.536443\pi$
$613$	− 49.1935i	− 1.98691i	0.114239	−	0.993453i	$-0.463557\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	− 46.0000i	− 1.84890i	−0.381308	−	0.924448i	$-0.624526\pi$
$619$	− 46.0000i	− 1.84890i	0.381308	−	0.924448i	$-0.375474\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−62.6099	−2.50841
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	58.1378i	2.30350i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−40.2492	−1.58236	−0.791180	−	0.611583i	$-0.790533\pi$
$647$	−40.2492	−1.58236	−0.791180	+	0.611583i	$0.790533\pi$
$648$	0	0
$649$	−28.0000	−1.09910
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.47214i	0.175008i	0.996164	+	0.0875041i	$0.0278891\pi$
$653$	4.47214i	0.175008i	−0.996164	+	0.0875041i	$0.972111\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	26.0000i	1.01282i	0.862294	+	0.506408i	$0.169027\pi$
$659$	26.0000i	1.01282i	−0.862294	+	0.506408i	$0.830973\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	49.1935i	1.89066i	0.326116	+	0.945330i	$0.394260\pi$
$677$	49.1935i	1.89066i	−0.326116	+	0.945330i	$0.605740\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	60.0000	2.28582
$690$	0	0
$691$	− 42.0000i	− 1.59776i	−0.601494	−	0.798878i	$-0.705427\pi$
$691$	− 42.0000i	− 1.59776i	0.601494	−	0.798878i	$-0.294573\pi$
$692$	0	0
$693$	26.8328i	1.01929i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	−26.8328	−1.01202
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−20.0000	−0.744839
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−4.47214	−0.165862	−0.0829312	−	0.996555i	$-0.526428\pi$
$727$	−4.47214	−0.165862	−0.0829312	+	0.996555i	$0.526428\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 49.1935i	− 1.81700i	−0.417881	−	0.908502i	$-0.637227\pi$
$733$	− 49.1935i	− 1.81700i	0.417881	−	0.908502i	$-0.362773\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 54.0000i	− 1.98642i	−0.116326	−	0.993211i	$-0.537112\pi$
$739$	− 54.0000i	− 1.98642i	0.116326	−	0.993211i	$-0.462888\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	40.2492	1.47660	0.738300	−	0.674472i	$-0.235629\pi$
$743$	40.2492	1.47660	0.738300	+	0.674472i	$0.235629\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	31.3050i	1.13780i	0.822407	+	0.568899i	$0.192630\pi$
$757$	31.3050i	1.13780i	−0.822407	+	0.568899i	$0.807370\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	62.6099	2.26071
$768$	0	0
$769$	54.0000	1.94729	0.973645	−	0.228069i	$-0.0732413\pi$
$769$	54.0000	1.94729	0.973645	+	0.228069i	$0.0732413\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	22.3607i	0.804258i	0.915583	+	0.402129i	$0.131730\pi$
$773$	22.3607i	0.804258i	−0.915583	+	0.402129i	$0.868270\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 12.0000i	− 0.429945i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 40.2492i	− 1.42570i	−0.701316	−	0.712850i	$-0.747404\pi$
$797$	− 40.2492i	− 1.42570i	0.701316	−	0.712850i	$-0.252596\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	42.0000	1.48400
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	− 2.00000i	− 0.0702295i	−0.999383	−	0.0351147i	$-0.988820\pi$
$811$	− 2.00000i	− 0.0702295i	0.999383	−	0.0351147i	$-0.0111797\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	− 60.0000i	− 2.09657i
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	−31.3050	−1.09122	−0.545611	−	0.838039i	$-0.683702\pi$
$823$	−31.3050	−1.09122	−0.545611	+	0.838039i	$0.683702\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−31.3050	−1.07565
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 20.0000i	− 0.685591i
$852$	0	0
$853$	58.1378i	1.99060i	0.0968435	+	0.995300i	$0.469125\pi$
$853$	58.1378i	1.99060i	−0.0968435	+	0.995300i	$0.530875\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	− 14.0000i	− 0.477674i	−0.971060	−	0.238837i	$-0.923234\pi$
$859$	− 14.0000i	− 0.477674i	0.971060	−	0.238837i	$-0.0767661\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	58.1378	1.97903	0.989516	−	0.144421i	$-0.0461320\pi$
$863$	58.1378	1.97903	0.989516	+	0.144421i	$0.0461320\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 58.1378i	− 1.96317i	−0.191018	−	0.981586i	$-0.561179\pi$
$877$	− 58.1378i	− 1.96317i	0.191018	−	0.981586i	$-0.438821\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	58.0000	1.95407	0.977035	−	0.213080i	$-0.0683494\pi$
$881$	58.0000	1.95407	0.977035	+	0.213080i	$0.0683494\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.47214	−0.150160	−0.0750798	−	0.997178i	$-0.523921\pi$
$887$	−4.47214	−0.150160	−0.0750798	+	0.997178i	$0.523921\pi$
$888$	0	0
$889$	100.000	3.35389
$890$	0	0
$891$	− 18.0000i	− 0.603023i
$892$	0	0
$893$	− 80.4984i	− 2.69378i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 98.3870i	− 3.24902i
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	13.4164	0.440653
$928$	0	0
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	− 78.0000i	− 2.55635i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	8.94427	0.291266
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	31.3050	1.00670	0.503350	−	0.864083i	$-0.332101\pi$
$967$	31.3050	1.00670	0.503350	+	0.864083i	$0.332101\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	62.0000i	1.98967i	0.101482	+	0.994837i	$0.467641\pi$
$971$	62.0000i	1.98967i	−0.101482	+	0.994837i	$0.532359\pi$
$972$	0	0
$973$	62.6099i	2.00718i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	− 28.0000i	− 0.894884i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	40.2492	1.28375	0.641875	−	0.766809i	$-0.278157\pi$
$983$	40.2492	1.28375	0.641875	+	0.766809i	$0.278157\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	49.1935i	1.55797i	0.627040	+	0.778987i	$0.284266\pi$
$997$	49.1935i	1.55797i	−0.627040	+	0.778987i	$0.715734\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.d.g.801.1		4
4.3	odd	2	inner	1600.2.d.g.801.4		4
5.2	odd	4		320.2.f.a.289.3	yes	4
5.3	odd	4		320.2.f.a.289.2	yes	4
5.4	even	2	inner	1600.2.d.g.801.3		4
8.3	odd	2	inner	1600.2.d.g.801.3		4
8.5	even	2	inner	1600.2.d.g.801.2		4
15.2	even	4		2880.2.d.e.289.1		4
15.8	even	4		2880.2.d.e.289.4		4
16.3	odd	4		6400.2.a.bj.1.1		2
16.5	even	4		6400.2.a.bj.1.2		2
16.11	odd	4		6400.2.a.bi.1.1		2
16.13	even	4		6400.2.a.bi.1.2		2
20.3	even	4		320.2.f.a.289.1	✓	4
20.7	even	4		320.2.f.a.289.4	yes	4
20.19	odd	2	inner	1600.2.d.g.801.2		4
40.3	even	4		320.2.f.a.289.3	yes	4
40.13	odd	4		320.2.f.a.289.4	yes	4
40.19	odd	2	CM	1600.2.d.g.801.1		4
40.27	even	4		320.2.f.a.289.2	yes	4
40.29	even	2	inner	1600.2.d.g.801.4		4
40.37	odd	4		320.2.f.a.289.1	✓	4
60.23	odd	4		2880.2.d.e.289.3		4
60.47	odd	4		2880.2.d.e.289.2		4
80.3	even	4		1280.2.c.c.769.2		2
80.13	odd	4		1280.2.c.b.769.2		2
80.19	odd	4		6400.2.a.bj.1.2		2
80.27	even	4		1280.2.c.b.769.2		2
80.29	even	4		6400.2.a.bi.1.1		2
80.37	odd	4		1280.2.c.c.769.2		2
80.43	even	4		1280.2.c.b.769.1		2
80.53	odd	4		1280.2.c.c.769.1		2
80.59	odd	4		6400.2.a.bi.1.2		2
80.67	even	4		1280.2.c.c.769.1		2
80.69	even	4		6400.2.a.bj.1.1		2
80.77	odd	4		1280.2.c.b.769.1		2
120.53	even	4		2880.2.d.e.289.2		4
120.77	even	4		2880.2.d.e.289.3		4
120.83	odd	4		2880.2.d.e.289.1		4
120.107	odd	4		2880.2.d.e.289.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.2.f.a.289.1	✓	4	20.3	even	4
320.2.f.a.289.1	✓	4	40.37	odd	4
320.2.f.a.289.2	yes	4	5.3	odd	4
320.2.f.a.289.2	yes	4	40.27	even	4
320.2.f.a.289.3	yes	4	5.2	odd	4
320.2.f.a.289.3	yes	4	40.3	even	4
320.2.f.a.289.4	yes	4	20.7	even	4
320.2.f.a.289.4	yes	4	40.13	odd	4
1280.2.c.b.769.1		2	80.43	even	4
1280.2.c.b.769.1		2	80.77	odd	4
1280.2.c.b.769.2		2	80.13	odd	4
1280.2.c.b.769.2		2	80.27	even	4
1280.2.c.c.769.1		2	80.53	odd	4
1280.2.c.c.769.1		2	80.67	even	4
1280.2.c.c.769.2		2	80.3	even	4
1280.2.c.c.769.2		2	80.37	odd	4
1600.2.d.g.801.1		4	1.1	even	1	trivial
1600.2.d.g.801.1		4	40.19	odd	2	CM
1600.2.d.g.801.2		4	8.5	even	2	inner
1600.2.d.g.801.2		4	20.19	odd	2	inner
1600.2.d.g.801.3		4	5.4	even	2	inner
1600.2.d.g.801.3		4	8.3	odd	2	inner
1600.2.d.g.801.4		4	4.3	odd	2	inner
1600.2.d.g.801.4		4	40.29	even	2	inner
2880.2.d.e.289.1		4	15.2	even	4
2880.2.d.e.289.1		4	120.83	odd	4
2880.2.d.e.289.2		4	60.47	odd	4
2880.2.d.e.289.2		4	120.53	even	4
2880.2.d.e.289.3		4	60.23	odd	4
2880.2.d.e.289.3		4	120.77	even	4
2880.2.d.e.289.4		4	15.8	even	4
2880.2.d.e.289.4		4	120.107	odd	4
6400.2.a.bi.1.1		2	16.11	odd	4
6400.2.a.bi.1.1		2	80.29	even	4
6400.2.a.bi.1.2		2	16.13	even	4
6400.2.a.bi.1.2		2	80.59	odd	4
6400.2.a.bj.1.1		2	16.3	odd	4
6400.2.a.bj.1.1		2	80.69	even	4
6400.2.a.bj.1.2		2	16.5	even	4
6400.2.a.bj.1.2		2	80.19	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.47214 q^{7} +3.00000 q^{9} -2.00000i q^{11} +4.47214i q^{13} -6.00000i q^{19} +4.47214 q^{23} -4.47214i q^{37} +2.00000 q^{41} +13.4164 q^{47} +13.0000 q^{49} -13.4164i q^{53} -14.0000i q^{59} -13.4164 q^{63} +8.94427i q^{77} +9.00000 q^{81} +14.0000 q^{89} -20.0000i q^{91} -6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9} + 8 q^{41} + 52 q^{49} + 36 q^{81} + 56 q^{89}+O(q^{100}) \) 4 * q + 12 * q^9 + 8 * q^41 + 52 * q^49 + 36 * q^81 + 56 * q^89

Introduction

L-functions

Modular forms

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